Methods for numerical conformal mapping

Journal Article · Tue Jul 15 00:00:00 EDT 1980 · J. Comput. Phys.; (United States)

DOI:https://doi.org/10.1016/0021-9991(80)90166-7· OSTI ID:5091457

Menikoff, R; Zemach, C

Nonlinear integral equations for the boundary functions which determine conformal transformations in two dimensions are developed and analyzed. One of these equations has a nonsingular logarithmic kernel and is especially well suited for numerical computations of conformal maps including those which deal with regions having highly distorted boundaries. Numerical procedures based on interspersed Gaussian quadrature for approximating the integrals and a Newton--Raphson technique to solve the resulting nonlinear algebraic equations are described. The Newton--Raphson iteration converges reliably with very crude initial approximations. Numerical examples are given for the mapping of a half-infinite region with periodic boundary onto a half plane, with up to nine-figure accuracy for values of the map function on the boundary and for its first derivatives. The examples include regions bounded by ''spike'' curves characteristic of Rayleigh--Taylor instability phenomena. A differential equation is derived which relates changes of the boundary. This is relevant to potential problems for regions with time-dependent boundaries. Further nonsingular integral formulas are derived for conformal mapping in a variety of geometries and for application to the boundary-value problems of potential theory.

• Research Organization: Theoretical Division, Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico 87545
• OSTI ID: 5091457
• Journal Information: J. Comput. Phys.; (United States), Vol. 36:3
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE
CONFORMAL MAPPING
INTEGRAL EQUATIONS
ITERATIVE METHODS
BOUNDARY-VALUE PROBLEMS
KERNELS
NONLINEAR PROBLEMS
NUMERICAL SOLUTION
RAYLEIGH-TAYLOR INSTABILITY
TIME DEPENDENCE
EQUATIONS
INSTABILITY
MAPPING
TOPOLOGICAL MAPPING
TRANSFORMATIONS
990200* - Mathematics & Computers